It remains to guarantee that: (1) [1]} can be parallelized, and (2) the marginal probabilities \(\mu _v^{\tau _{\mathcal {M} \setminus \lbrace v\rbrace }}\)  can be calculated in parallel.
For (1), generally [1]} samples from a \(\mu _S^{\tau _T}\) , where \(S,T\subset V\)  are disjoint,
by: first constructing the connected components that intersect \(S\)  in the formula of clauses not yet satisfied by \({\tau _T}\) ,
and then a rejection samplings through independent trials applied on these components,
where the former can be parallelized by a connected component algorithm such as [3]} and the latter is trivially parallelizable.
For (2), due to Chernoff-Hoeffding, the marginal probabilities \(\mu _v^{\tau _{\mathcal {M} \setminus \lbrace v\rbrace }}\)  can be estimated within \(1/\mathrm {poly}(n)\)  additive error with \(1-1/\mathrm {poly}(n)\)  confidence, by independently (thus parallel) repeating [1]} for \(\mathrm {poly}(n)\)  times.
Overall, it only introduces \(1/\mathrm {poly}(n)\)  total variation error into the final output.
